From Schanuel’s Conjecture to Shapiro’s Conjecture P. D’Aquino,∗ A. Macintyre †and G. Terzo‡ October 29, 2018 Abstract In this paper we prove Shapiro’s 1958 Conjecture on exponential polyno- mials, assuming Schanuel’s Conjecture. 1 Introduction We work with exponential polynomial functions on C of the form µ1z µN z f(z)= λ1e + ... + λN e . (1) The set of such functions forms a ring E under the usual addition and multiplica- tion. We normally refer to exponential polynomial functions simply as exponential polynomials. In (1), we assume without loss of generality that the exponents µ’s are distinct, and that the coefficients λ’s are nonzero, unless f is the zero polynomial. In 1974 during the Janos Bolyai Society Colloquium on Number Theory, H. arXiv:1206.6747v1 [math.NT] 28 Jun 2012 L. Montgomery mentioned the following conjecture, which he attributed to H. S. Shapiro [15]: Shapiro’s Conjecture: If f and g are two exponential polynomials in E with infinitely many common roots, then there exists an exponential polynomial h in E such that ∗Department of Mathematics, Seconda Universit`adi Napoli, Via Vivaldi 43, 81100 Caserta, [email protected] †School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, [email protected] ‡Department of Mathematics, Seconda Universit`adi Napoli, Via Vivaldi 43, 81100 Caserta, [email protected] 1 h is a common divisor of f and g in the ring E, and h has infinitely many zeros in C. Montgomery pointed out, via an example given in [5], that the problem was not likely to yield easily to any classical approximation argument. It turns out that Shapiro’s Conjecture is naturally connected to Schanuel’s Con- jecture in Transcendence Theory. Schanuel’s Conjecture: Let λ1,...,λn ∈ C. Then the transcendence degree of λ1 λn Q(λ1,...,λn, e ,...,e ) over Q is greater or equal than the linear dimension of λ1,...,λn over Q. Schanuel’s Conjecture has played a crucial role in exponential algebra (see [10], [16]), and in the model theory of exponential fields (see [11], [17], [12]). In [13] Ritt obtained a factorization theory for exponential polynomials in E. Subsequently, his ideas have been developed, and his results have been extended to more general exponential polynomials over C, see [7], [9], [3], [4]. In [2], these ideas have been put in the much broader context of general exponential polynomials (with any iteration of exponentiation) over an algebraically closed field of characteristic 0 with an exponentiation. In this paper we will study the Shapiro Conjecture in a context more general than that of the complex field. We will be working over an algebraically closed field of characteristic 0, with an exponential function, and having an infinite cyclic group of periods, whose exponential is surjective onto the multiplicative group. The class of such fields includes the very important fields introduced by Zilber (see [17] for the basic notions). The preceding assumptions play a minor role in our work on Shapiro’s Conjecture. Of crucial importance is our further assumption, true for Zilber’s fields, but unproved for the complex field, that we work with exponential fields satisfying Schanuel’s Conjecture. One should note that in an exponential field satisfying the above assumptions (even without the surjectivity of the exponential onto the multiplicative group) the two element set of generators of the periods is first-order definable [8], the sine and cosine function are unambiguously defined, and the two element set consisting of the quotients of the period generators by twice a square root of −1 is definable (the set does not depend on which root is chosen). In C this would define the set {π, −π}. Finally, we can define the one element subset consisting of the element x such that sin(x/2) = 1. In a general field satisfying our assumptions, we call this element π. In section 2 we review the basic ideas of Ritt’s factorization theory for expo- nential polynomials. His main theorem allows us to break the proof of Shapiro’s 2 conjecture into two cases. One case was already done by van der Poorten and Tijde- man [5] for simple polynomials (in Ritt’s sense) over C, without any use of Schanuel’s Conjecture. In section 3, we modify that argument so as to apply to fields satisfying all the assumptions given above with the exception of Schanuel’s Conjecture. Section 4 explains recent work of Bombieri, Masser and Zannier [1] on anomalous subvarieties of powers of the multiplicative group. The main result of this paper is in section 5 where a positive solution to Shapiro’s Conjecture is obtained for the remaining case of irreducible exponential polynomials, assuming Schanuel’s Conjecture, using the work of Bombieri, Masser and Zannier, and work of Evertse, Schlikewei and Schmidt on linear functions of elements of multiplicative groups of finite rank. We feel obliged to make a philosophical remark about the use of Schanuel’s Conjecture to “settle” a conjecture which emerged from complex analysis. A very distinguished number theorist has remarked that if one assumes Schanuel’s Conjec- ture one can prove anything. The sense of this is clear if one restricts “anything” to refer to statements in transcendence theory. In that domain the Conjecture is almost a machine, leading one mechanically to “proofs” of any plausible conjec- tures about algebraic relations between complex numbers and their exponentials. We suspect that there is also a common intuition that statements about common zeros of exponential poynomials should be related to statements about the tran- scendence theory of the exponential function. However, the original motivation for Shapiro’s Conjecture clearly comes from reflection on distribution of zeros of indi- vidual exponential polynomials, and predates Schanuel’s Conjecture. Moreover, our argument involves combinatorial considerations not previously connected to routine applications of Schanuel’s Conjecture. 2 Factorization Theory We briefly review the main ideas in Ritt’s factorization for exponential polynomials in E. Most of the theory adapts to the much more general context of the ring of exponential polynomials over an algebraically closed field of characteristic 0 with an exponential function (see [2]). The fundamental idea due to Ritt was to transform problems of factorization of exponential polynomials to those of factorization of classical multivariate polyno- mials in the extended category of polynomials in fractional powers of the variables. This brings in the notion of power irreducible multivariate polynomial explained below. In general, if we consider an irreducible polynomial Q(x1,...,xn) it can happen 3 q1 qn that for some positive integers q1,...,qn the polynomial Q(x1 ,...,xn ) is reducible. q1 qn If there exists no sequence q1,...,qn of positive integers such that Q(x1 ,...,xn ) is reducible we will refer to Q as a power irreducible polynomial. We briefly review how to associate a classical polynomial in one or more variables to an exponential polynomial in E. We collect some basic definitions and results. Fact. The units in the ring E are the products of nonzero constants and eαz for constant α ∈ C. Definition 2.1. An element f in E is irreducible, if there are no non-units g and h in E such that f = gh. N µiz Definition 2.2. Let f = i=1 αie be an exponential polynomial. The support of f, denoted by supp(f), isP the Q-space generated by µ1,...,µN . Definition 2.3. An exponential polynomial f(z) of E is simple if dim supp(f)=1. It is easily seen that, up to a unit, a simple exponential polynomial is a poly- nomial in eµz, for some µ ∈ C. An example of a simple exponential polynomial is e2πiz − e−2πiz g(z)= = sin(2πz). 2i Remark 2.4. A simple exponential polynomial factorizes, up to units, into a finite product of factors of the form 1 − αeµz, where α,µ ∈ C. This simply uses the fact that the complex field is algebraically closed. If fractional powers of the variables are allowed then a simple exponential polynomial may have infinitely many factors, µz e.g. 1 − αe k , for each k ∈ N,k =6 0. µ1z µN z Let f(z)= λ1e + ... + λN e where λi and µi are complex numbers, and let βj z β1,...βr be a Z-basis of the additive group generated by the µi’s. Let Yj = e , with j =1, . , r. If each µi is expressed in terms of the βj’s we have that f is transformed into a classical Laurent polynomial Q over C in the variables Y1,...,Yr. The best way to think of Q is as a function on the product of r copies of the multiplicative group variety. Remember that the Y ’s are exponentials and so take value in the multiplicative group. More prosaically, one can write Q as a product of a polynomial in the Y ’s and a quotient of monomials in the Y ’s. Clearly, any factorization of f determines a factorization of Q(Y1,...,Yr). Ritt saw the relevance, in terms of factorization theory, of understanding the ways in which an irreducible polynomial Q(Y1,...,Yr) can become reducible once the vari- ables are replaced by their powers. It is a fundamental problem to determine the set 4 of integer r-tuples q1,...,qr for which the reducibility occurs. Ritt gave a uniform q1 qr bound for the number of irreducible factors of Q(Y1 ,...,Yr ), depending only on the degree of Q. For the factorization theorem of Ritt the following lemma is crucial. N µiz M mj z Lemma 2.5.